arXiv CSopen access

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression

arXiv:2609.04995v1 [cs.LG] 4 Sep 2026

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression Juncheng Zhou∗

Jiaxi Lu∗

Weijing Zeng

School of Cyber Science and Engineering, Wuhan University Wuhan, China [email protected]

School of Cyber Science and Engineering, Wuhan University Wuhan, China [email protected]

School of Mathematics and Statistics, Wuhan University Wuhan, China [email protected]

Zhong Li

Hao Qi

Jingsong Cui✉

School of Synthetic Biology and Biomanufacturing, Tianjin University Tianjin, China [email protected]

School of Synthetic Biology and Biomanufacturing, Tianjin University Tianjin, China [email protected]

School of Cyber Science and Engineering, Wuhan University Wuhan, China [email protected]

Abstract Deep Imbalanced Regression (DIR) is pervasive in continuous prediction tasks across diverse modalities, such as age estimation, depth prediction, and protein mutation activity prediction, where labelscarce tail samples often carry higher practical value. However, most existing methods still learn deterministic point mappings under mean squared error or its simple variants, implicitly assuming a uniform uncertainty level across all samples and thereby overlooking the instance-wise heteroscedasticity that is widespread in long-tailed data. We further point out that even heteroscedastic negative log-likelihood suffers from a gradient coupling issue, which, under DIR scenarios, weakens the learning signal of hard tail samples and leads to optimization inertia as well as tail underfitting. To address this, we propose DUO, an uncertainty-aware longtailed regression framework. Specifically, the proposed method models the regression target as a conditional Gaussian distribution to explicitly characterize instance-level predictive uncertainty, and transforms uncertainty into a dynamic enhancement signal for tail samples through decoupled mean-variance optimization. Furthermore, we design a distribution-guided contrastive learning mechanism that adaptively constructs positive and negative pairs based on the overlap between sample distributions, thereby alleviating feature looseness and cross-label semantic entanglement. Across visual and biological DIR benchmarks, DUO achieves the best few-shot bMAE and GM on IMDB-WIKI-DIR, AgeDB-DIR, and AAV2-DIR while remaining competitive on few-shot MAE.

CCS Concepts • Computing methodologies → Supervised learning; Learning latent representations.

Keywords Deep Imbalanced Regression, Long-Tailed Regression, Predictive Uncertainty, Contrastive Learning, Decoupled Optimization ∗ Juncheng Zhou and Jiaxi Lu contributed equally to this work.

Accepted at the 34th ACM International Conference on Multimedia (ACM MM 2026). The Version of Record will be available at https://doi.org/10.1145/3767308.3835898.

(a) Uncertainty Variation from Image Quality High-Quality Image Tight Distribution

Low-Quality Image Tight Distribution

Age Uncertainty Range ≈ 2 (33 – 37 years)

Age Uncertainty Range ≈ 5 (30 – 40 years)

30

33

35

37

40

Age (Years)

(b) Uncertainty Variation from Data Density

33

35

Common Age (Dense Data) Tight Distribution

Rare Age (Sparse Data) Wide Distribution

Age Uncertainty Range ≈ 2 (33 – 37 years)

Age Uncertainty Range ≈ 10 ( 70– 90 years)

37

70

80

90

Age (Years)

Figure 1: Illustration of instance-level heteroscedasticity. (a) For the same age label, predictive uncertainty increases as image quality degrades. (b) Predictive uncertainty also increases in sparse label regions, even for high-quality samples.

1

Introduction

Deep Imbalanced Regression (DIR) is prevalent across numerous real-world continuous prediction tasks, ranging from facial age and depth estimation in multimedia systems to protein mutation activity prediction in computational biology. Such tasks typically exhibit a prominent long-tailed distribution, wherein a majority of the labels dominate while a minority remain extremely scarce. This data imbalance heavily biases optimization toward the massive, easy-to-fit majority ("Many"). Consequently, models suffer from severe prediction bias against the scarce tail ("Few"), leading to systematically less reliable predictions on underrepresented samples. Tail categories can nevertheless have substantial collective influence[40], and individual rare cases may be consequential in application-specific settings.

Juncheng Zhou, Jiaxi Lu, Weijing Zeng, Zhong Li, Hao Qi, & Jingsong Cui

For instance, Deep Mutational Scanning (DMS)[11, 12] of AAV2 capsid proteins[27] provides large fitness landscapes with pronounced imbalance. Bryant et al. [6] focus on a 28-residue AAV2 segment (VP1 positions 561–588) that overlaps known antibodybinding sites and generate highly diverse variants that remain viable for packaging. In our processed AAV2-DIR data, the fraction labeled active drops from approximately 10% to 0.3% as the mutation count increases (≥ 6), approaching zero beyond 21 mutations. This scarcity yields a highly skewed target distribution, making reliable prediction in the sparse region important for identifying unusual viable variants. To alleviate long-tailed bias, existing research extensively explores data re-weighting, re-sampling, and architecture design[4, 13, 32, 34, 38]. However, most methods still optimize using Mean Squared Error (MSE) or its simple variants as the regression loss, which limits their ability to reflect sample-wise uncertainty. From a statistical modeling perspective, MSE entails an implicit assumption of global homoscedasticity, which presumes that all samples share an identical level of confidence (or uncertainty) during prediction. Real-world data often exhibit instance-level heteroscedasticity, where predictive uncertainty varies with both image quality and label density, which is particularly relevant to responsible multimedia systems that require reliable predictions under heterogeneous conditions. As shown in Figure 1(a), even at the same age label (e.g., 35 years old), a clear image yields a sharp distribution with low uncertainty (33-37 years, 𝜎 ≈ 2), whereas a blurry image produces a much wider distribution (30-40 years, 𝜎 ≈ 5). Figure 1(b) further shows that uncertainty also depends on label density: a commonage sample can remain highly confident (33-37 years, 𝜎 ≈ 2), while a high-quality sample from a sparse tail region (e.g., age 80) may still exhibit much higher uncertainty (70-90 years, 𝜎 ≈ 10). These observations reveal the limitation of MSE, which implicitly assumes uniform uncertainty across samples and may obscure reliability disparities across underrepresented cases. While mainstream DIR methods have made significant progress, their reliance on homoscedastic assumptions often introduces two distinct challenges. At the optimization level, highly uncertain tail samples can disrupt gradient updates, potentially leading to underfitting in scarce regions and reduced reliability on rare samples. At the representation level, treating heterogeneous samples uniformly tends to entangle cross-label semantics and degrade intra-label feature compactness. Consequently, shifting from deterministic point mapping to modeling samples as conditional probability distributions has emerged as a promising paradigm to capture data heterogeneity and provide more transparent uncertainty estimates. Following this paradigm, prior works have explored heteroscedastic modeling using the standard Negative Log-Likelihood (NLL) loss. However, jointly optimizing the mean and variance in NLL can compromise mean fitting (Stirn et al. [33]). Under long-tailed distributions, this issue is drastically exacerbated, inducing "optimization inertia." Recent uncertainty-aware DIR approaches instead adopt probabilistic smoothing or multi-expert aggregation[17, 37], but these mechanisms add modeling or inference complexity. To address these challenges, we propose DUO, an uncertaintyguided long-tailed regression framework. By modeling the prediction target as a conditional Gaussian distribution, DUO explicitly captures instance-level uncertainty through two core designs,

thereby better characterizing sample-wise predictive reliability. First, a mean-variance decoupled optimization mechanism utilizes gradient detachment to adaptively weight gradients, mitigating the gradient degradation issue in traditional likelihood losses. Second, a distribution-guided contrastive learning module employs the Bhattacharyya coefficient to quantify distribution overlap, thereby adaptively constructing contrastive pairs to alleviate the feature looseness and semantic entanglement of tail samples. In summary, our main contributions are threefold: (1) we formulate long-tailed regression as a conditional Gaussian problem to capture instance-level uncertainty, and analyze how mean–variance gradient coupling in NLL impairs optimization in DIR; (2) we propose the DUO loss function, which incorporates gradient detachment for computationally decoupled optimization and distributionguided contrastive learning; and (3) we validate DUO across multimedia and biological DIR benchmarks, demonstrating consistent gains on imbalance-aware few-shot metrics.

2 Related Work 2.1 Imbalanced Classification Existing imbalanced classification methods are broadly divided into two mainstream paradigms: reweighting/resampling-based approaches (RW/RS) and decision boundary and prior calibrationbased approaches (DBC/PC). RW/RS methods[8, 14, 18] mitigate long-tailed optimization bias by adjusting the gradient contributions of different classes/samples during training, with representative works including cost-sensitive learning based on inverse class frequency, Class-Balanced Loss that avoids over-amplification of tail classes via effective sample number (Cui et al.[10]), and Focal Loss that alleviates gradient domination by focusing on hard samples (Lin et al.[22]). DBC/PC methods[39, 41] tackle imbalance from the perspective of classification boundary geometry or label prior mismatch, such as LDAM with class-dependent margin optimization (Cao et al.[7]), the decoupled training paradigm that separates representation learning and classifier rebalancing (Kang et al.[20]), and Logit Adjustment(Menon et al.[23]) and Balanced Softmax(Ren et al.[28]) that correct prior mismatch at the logit level. However, all above methods are built on the structural assumptions of discrete classes, class priors, or classification boundaries. This inherent discrete label dependency makes them difficult to directly transfer to imbalanced regression tasks with continuous label spaces, where the coupling between sample scarcity, predictive uncertainty and feature geometry remains to be systematically modeled.

2.2

Imbalanced Regression

Compared with imbalanced classification, imbalanced regression faces greater challenges due to the continuous nature of label space and the absence of explicit class boundaries. Sample scarcity in IR no longer manifests as class frequency imbalance, but as highly skewed label distributions with severe sparsity and noise in tail regions (Yang et al.[38]), making most classification methods inapplicable. Existing IR approaches typically intervene at the input, feature, or output level, but most operate on only a single level and fail to model the coupling between sample scarcity and predictive uncertainty.

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression

Input-level methods mitigate label imbalance by adjusting training data distribution or sample weights directly[4, 5, 8, 34]. Early works like SMOTER[34] and SMOGN[4] define rare regions in continuous label space, and perform oversampling for tail samples while undersampling head ones. However, these methods only adjust the data distribution at the input stage, without addressing the core issues of feature collapse and prediction bias in tail regions during model optimization. Feature-level methods preserve continuous label relationships in representation space via structural constraints on the feature space. The seminal DIR work proposes Label Distribution Smoothing (LDS) and Feature Distribution Smoothing (FDS), which alleviate tail feature collapse by sharing statistical information within local label neighborhoods (Yang et al.[38]). Building on this, RankSim[13] introduces global geometric constraints by aligning the relative ordering between label and feature spaces, and ConR[21] further leverages contrastive learning to penalize label-feature inconsistency and prevent minority sample collapse. While these methods effectively model feature geometry for continuous labels, they do not incorporate predictive uncertainty into the feature learning process, leaving the coupling between sample scarcity and uncertainty under-explored. Output-level methods mainly mitigate tail prediction bias by modifying loss functions or predictive distributions. DenseWeight[32] estimates target rarity via kernel density estimation, while LDSbased reweighting smooths empirical label densities before assigning sample weights[38]. Balanced MSE (BMSE)[29] corrects the discrepancy between training and target priors, and Dist Loss[25] jointly penalizes sample-wise errors and prediction–label distribution distance. These methods correct prediction bias but do not use uncertainty to guide upstream representation learning. Recent works have introduced uncertainty into imbalanced regression: VIR [37] uses probabilistic neighborhood smoothing for long-tailed uncertainty quantification, while UVOTE [17] uses predictive uncertainty to select among ensemble experts. These methods model uncertainty primarily at the output or predictivedistribution level rather than using it to construct representationspace relations, leaving uncertainty-guided feature learning underexplored. In summary, existing IR methods generally alleviate label sparsityinduced bias from a single perspective, but fail to explicitly model the coupling among sample scarcity, predictive uncertainty, and feature geometry in continuous label spaces. Motivated by this gap, we propose DUO, which explicitly models sample-level uncertainty and uses it as a unified guidance signal for both optimization and representation learning, thereby improving tail sample modeling in long-tailed regression.

However, DIR data is inherently heteroscedastic: tail samples exhibit significantly higher uncertainty and noise compared to well-represented head samples. Ignoring this variance prevents the model from dynamically adapting to sample difficulty, fundamentally limiting its generalization on the tail.

3.2

Probabilistic Regression Framework

To overcome the limitations of deterministic mappings, we reformulate the regression task as modeling the conditional probability distribution 𝑝 (𝑦|𝑥)[2]. Specifically, we draw inspiration from classical classification, where learning a conditional probability distribution is the standard paradigm. Probabilistic Modeling for Continuous Space In standard classification, the model essentially learns a discrete conditional probability distribution. Typically, it outputs a normalized probability vector (i.e., via Softmax) and makes decisions based on the Maximum A Posteriori (MAP) principle [3]: 𝑦ˆ = arg max 𝑝 (𝑦 = 𝑘 |𝑥)

(1)

𝑘

This indicates that the core of classification lies in identifying the category with the highest confidence. From an isomorphic standpoint, regression should not be confined to predicting a single scalar; rather, it should be conceptualized as locating the point of maximum probability density within a continuous label space. To this end, we model the target conditional distribution as a Gaussian:
𝑝 (𝑦|𝑥; 𝜃 ) = N 𝑦; 𝜇 (𝑥), 𝜎 2 (𝑥) (2)

3 Method 3.1 Problem Formulation

The choice of a Gaussian formulation is fundamentally principled. From an information-theoretic perspective, given the constraints of the first moment (mean) and second moment (variance), the Gaussian distribution uniquely maximizes differential entropy[16], thereby introducing the minimum inductive bias into our model (we provide the rigorous mathematical proof of this maximum entropy optimality in Appendix A). Under this framework, the prediction 𝑦ˆ corresponds to the mode (which is also the mean 𝜇 (𝑥)) of the distribution, mathematically unifying classification and regression. Instance-Level Heteroscedasticity Modeling The necessity of explicitly modeling the second moment 𝜎 2 (𝑥) stems from correcting the bias inherent in traditional methods. Although standard MSE regression is equivalent to maximizing a Gaussian likelihood, its implicit homoscedasticity assumption forces all samples to share a fixed confidence, which contradicts the intrinsic heteroscedasticity of long-tailed data. Furthermore, since aleatoric uncertainty inherently arises from the input 𝑥 (e.g., occlusion or noise) rather than the target label 𝑦, formulating 𝜎 2 (𝑥) at the instance level is statistically more rigorous than relying on a class-level prior. This instance-wise formulation enables the model to dynamically quantify the specific ambiguity of each sample, providing a robust statistical foundation for regression.

DIR aims to learn a robust model from a long-tailed dataset D = 𝑁 , where 𝑝 (𝑦 {(𝑥𝑖 , 𝑦𝑖 )}𝑖=1 head ) ≫ 𝑝 (𝑦 tail ). Conventional methods learn a deterministic mapping 𝑓 : X → R by minimizing the Mean Squared Error (MSE). Statistically, MSE models solely the first moment (mean) of 𝑝 (𝑦|𝑥), implicitly assuming homoscedasticity (i.e., constant variance across all samples).

To instantiate this probabilistic framework, we minimally extend standard architectures. A shared backbone 𝑓enc (e.g., ResNet-50[15]) first extracts features 𝑧 = 𝑓enc (𝑥) ∈ R𝐷 . We then employ a dualbranch structure with a mean prediction head H𝜇 and a parallel

3.3

Network Architecture

Juncheng Zhou, Jiaxi Lu, Weijing Zeng, Zhong Li, Hao Qi, & Jingsong Cui

Network Architecture and Probabilistic Framework

Decoupled Mean-Variance Optimization

Gradient Flow Gradient Backpropagation

Mean Loss ℒ𝑀𝑒𝑎𝑛 1 ^ ^ ℒ 𝑀𝑒𝑎𝑛 = (1 + 𝑤(𝑦) ⋅ sg[𝜎 (𝑥)]) ⋅ ‖𝑦 − 𝜇 (𝑥)‖2 2

Z ^

𝑦𝑖

^

Uncertainty Head

𝜎𝑖2

SG

Variance Loss ℒ𝑉𝑎𝑟 ^

^ ^

𝑝(𝑦|𝑥𝑖 ) = 𝒩(𝑦; 𝜇𝑖 , 𝜎𝑖2)

ℒ 𝑉𝑎𝑟 = 𝛽 ⋅ ((

Distribution-based Pair Assignment

(𝑦 − sg[𝜇 (𝑥)])2 ^

2𝜎 2 (𝑥)

1 ^ ) + log( 𝜎 2 (𝑥))) 2

𝒩𝑖 = 𝒩

^ 𝑦𝑖 𝜎𝑖2

𝒩𝑗 = 𝒩

^ 𝑦𝑗 𝜎𝑗2

𝑠𝑖𝑗 ≥ 𝜏𝑜𝑣𝑒𝑟𝑙𝑎𝑝

Gradient Detachment

SG

Distribution-Guided Contrastive Learning Epoch 1 - k

Pair Distributions

Shared Backbone

Shared Backbone

Modeling

Uncertainty Head

Mean Head

Mean Head

^

𝜇𝑖

𝑠𝑖𝑗 < 𝜏𝑜𝑣𝑒𝑟𝑙𝑎𝑝

Training Phase Control

Warm-Up Stage Contrastive OFF

Epoch k+ After Warm-Up Contrastive ON

Align Loss ℒ𝐴𝑙𝑖𝑔𝑛

BC Score 𝑠𝑖𝑗 = BC 𝒩𝑖 𝒩𝑗

Negative Pair

Positive Pair

Figure 2: Illustration of the proposed DUO framework. Top-left: a probabilistic regression architecture that models each sample as a Gaussian distribution with predicted mean and uncertainty. Top-middle and top-right: a decoupled mean-variance optimization strategy with stop-gradient, which separates mean learning from variance fitting and explicitly controls gradient flow. Bottom-left: distribution-based pair assignment using the Bhattacharyya coefficient to determine whether two samples form a positive or negative pair according to their distribution overlap. Bottom-right: a distribution-guided contrastive learning module, which is activated after warm-up to refine the feature space for DIR. uncertainty head H𝜎 . A positive parameterization 𝑔+ ensures a valid standard deviation: 𝜇ˆ (𝑥) = 𝑊𝜇 𝑧 + 𝑏 𝜇 , 𝜎ˆ (𝑥) = 𝑔+ (𝑊𝜎 𝑧 + 𝑏𝜎 ),

𝑔+ (·) > 0.

(3)

This design seamlessly parameterizes 𝑝 (𝑦|𝑥) = N (𝑦; 𝜇ˆ (𝑥), 𝜎ˆ 2 (𝑥)) with negligible computational overhead.

3.4

Decoupled Mean-Variance Optimization

To train the model robustly on long-tailed distributions, we first analyze the optimization failure of the standard Negative LogLikelihood (NLL) loss. We then propose a decoupled objective using the stop-gradient operator. 3.4.1 Gradient Coupling in NLL Loss. For heteroscedastic Gaussian regression, the standard objective is the NLL loss LNLL [26]: LNLL =

ˆ 2 (𝑦 − 𝜇) 1 log 𝜎ˆ 2 + 2 2𝜎ˆ 2

(4)

ˆ 2 𝜕LNLL 𝜎ˆ 2 − (𝑦 − 𝜇) = 𝜕𝜎ˆ 2 2𝜎ˆ 4

3.4.2 Decoupled Objective. To resolve this gradient coupling, DUO employs the stop-gradient (sg[·]) operator[9, 35] to computaˆ tionally decouple the optimization of 𝜇ˆ and 𝜎. Mean Loss To address the gradient vanishing in NLL, we utilize uncertainty to amplify rather than suppress the gradient. The mean loss is defined as: 1 L𝑀𝑒𝑎𝑛 = (1 + 𝑤 (𝑦) · sg[𝜎ˆ (𝑥)]) · ∥𝑦 − 𝜇ˆ (𝑥) ∥ 2 | {z } 2

(6)

𝛾 :Dynamic Difficulty Weight

To investigate its behavior under long-tailed distributions, we examine the gradient of this loss function with respect to the mean prediction 𝜇ˆ and variance 𝜎ˆ 2 : 𝜕LNLL 1 ˆ = − 2 (𝑦 − 𝜇), 𝜕 𝜇ˆ 𝜎ˆ

At this equilibrium, the mean gradient magnitude behaves as ˆ → 0, leading to vanishing updates for large∥∇𝜇ˆ L ∥ ≈ 1/|𝑦 − 𝜇| error samples. In the DIR setting, tail samples are typically associated with larger residuals, which further amplifies this effect: samples that require the most correction receive the weakest learning signal. Consequently, the model tends to reduce the loss by increasing the predicted variance rather than improving the mean prediction.

(5)

Examining the gradients reveals a gradient coupling issue: the ˆ 𝜎ˆ 2 is scaled by 1/𝜎ˆ 2 . As proven mean gradient ∇𝜇ˆ L = −(𝑦 − 𝜇)/ in Appendix B, NLL optimization drives 𝜎ˆ 2 toward the squared ˆ 2 for hard samples (under both gradient descent and residual (𝑦 − 𝜇) Newton’s method).

where the continuous label space is discretized into bins using the training set, and 𝑤 (𝑦) is the inverse empirical frequency of the bin containing 𝑦. ˆ Here, = −𝛾 · (𝑦 − 𝜇). The gradient with respect to 𝜇ˆ is 𝜕 L𝜕Mean 𝜇ˆ 𝑤 (𝑦) captures global label scarcity, while 𝜎ˆ (𝑥) captures instancelevel difficulty. The latter provides a unidirectional forward weight but, due to sg[·], receives no gradient from LMean . Thus, forward numerical dependence does not reintroduce backward optimization coupling. Variance Loss

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression

Figure 3: Overview of label distributions in the training sets for the AgeDB-DIR, IMDB-WIKI-DIR, Nydu2-DIR and AAV2-DIR datasets. Since ground-truth variance is unavailable, learning 𝜎ˆ 2 is unsupervised. To tightly approximate the prediction error, we design the decoupled variance loss: L𝑉 𝑎𝑟 =

(𝑦 − sg[ 𝜇ˆ (𝑥)]) 2 1 + log 𝜎ˆ 2 (𝑥) 2𝜎ˆ 2 (𝑥) 2

(7)

This objective balances residual fitting and variance regularization. Specifically, the log 𝜎ˆ 2 term penalizes unnecessarily large variance estimates, while the inverse-variance term penalizes underestimated variance when the mean prediction error is large. Because sg[·] stops gradients through 𝜇ˆ (𝑥), the variance branch is optimized against a fixed residual target. Setting the gradient with respect to 𝜎ˆ 2 to zero yields the stationary point 𝜎ˆ 2 (𝑥) = (𝑦 − 𝜇ˆ (𝑥)) 2 . Appendix C further proves that this point is the unique global minimizer of L𝑉 𝑎𝑟 in the linear variance space, and that under log-variance parameterization the optimization converges exponentially in the KL/Bregman sense. Therefore, 𝜎ˆ serves as an implicit proxy for the empirical prediction residual.

the overlap integral of two probability density functions: ∫ √︁ BC(𝑖, 𝑗) = 𝑝 (𝑦|𝑥𝑖 ) · 𝑝 (𝑦|𝑥 𝑗 ) 𝑑𝑦

Substituting the Gaussian proxy distributions into the overlap integral yields the following closed-form solution, whose full derivation is provided in Appendix D: ! 2 2 1 (𝑦𝑖 − 𝑦 𝑗 ) 2 1 𝜎ˆ𝑖 + 𝜎ˆ 𝑗 (10) − ln BC(𝑖, 𝑗) = exp − 4 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 2 2𝜎ˆ𝑖 𝜎ˆ 𝑗 Essentially, this metric introduces an adaptive semantic tolerance. ˆ BC remains high even For high-uncertainty tail samples (large 𝜎), if the label distance |𝑦𝑖 − 𝑦 𝑗 | is large, provided the distributions overlap. This allows hard samples to interact with a broader set of semantic neighbors during manifold alignment. Based on this probabilistic similarity, we dynamically define the positive and negative sample sets respectively as P𝑖 = { 𝑗 | BC(𝑖, 𝑗) ≥ 𝜏overlap } N𝑖 = { 𝑗 | BC(𝑖, 𝑗) < 𝜏overlap }

3.5

Distribution-Guided Contrastive Learning

While the decoupled loss optimizes output-space predictions, longtailed data still severely degrades feature representations, often leading to feature entanglement in tail classes. Standard contrastive learning uses rigid label distance thresholds (e.g., |𝑦𝑖 − 𝑦 𝑗 | < 𝜖) to define positive pairs, ignoring the instance-level variance. To address this, we propose a distribution-guided contrastive learning module that dynamically shapes the feature manifold based on the geometric overlap of predicted distributions. 3.5.1 Pair Assignment via Distribution Overlap. To measure semantic similarity in the feature space, we construct a label-anchored proxy distribution. Since ground-truth variance is unavailable, we model each sample 𝑥𝑖 as a Gaussian proxy distribution centered at its ground-truth label 𝑦𝑖 with predicted variance 𝜎ˆ𝑖2 : 𝑝 (𝑦 | 𝑥𝑖 ) = N (𝑦; 𝑦𝑖 , 𝜎ˆ𝑖2 )

In this way, 𝜎ˆ𝑖 captures the uncertainty range of sample 𝑥𝑖 in label space, while the ground-truth label 𝑦𝑖 provides a stable semantic anchor for pair assignment. We quantify the geometric overlap between two proxy distributions using the Bhattacharyya Coefficient (BC)[1, 19], defined as

(11)

thereby overcoming the limitations of rigid distance-based thresholds. 3.5.2 Distance-Aware Contrastive Objective. To correct feature entanglement where samples with large label discrepancies are erroneously clustered, we design a distance-aware InfoNCE loss L𝐴𝑙𝑖𝑔𝑛 to geometrically constrain the backbone features 𝑧:

L𝐴𝑙𝑖𝑔𝑛 = −

∑︁ 𝑖

Í log Í

𝑗 ∈ P𝑖 𝑒

𝑗 ∈ P𝑖 𝑒

cos(𝑧𝑖 ,𝑧 𝑗 )/𝜏 +

cos(𝑧𝑖 ,𝑧 𝑗 )/𝜏

Í

𝑘 ∈ N𝑖 W𝑝𝑢𝑠ℎ (𝑖, 𝑘) · 𝑒

cos(𝑧𝑖 ,𝑧𝑘 )/𝜏

(12) To impose stronger repulsion on semantically incompatible yet feature-similar negatives, we introduce a dynamic penalty weight defined as the inverse Bhattacharyya overlap: W𝑝𝑢𝑠ℎ (𝑖, 𝑘) = 𝑤 (𝑦𝑖 ) · BC(𝑖, 𝑘) −1

(8)

(9)

(13)

Here, 𝑤 (𝑦𝑖 ) amplifies the gradient contribution of tail samples, while the inverse distribution overlap BC(𝑖, 𝑘) −1 heavily penalizes hard negatives with vanishing semantic overlaps. Appendix E shows that this weighting induces exponentially stronger repulsion for erroneously entangled negatives, helping restore an ordered manifold topology.

Juncheng Zhou, Jiaxi Lu, Weijing Zeng, Zhong Li, Hao Qi, & Jingsong Cui

During the first 𝑇𝑤 warm-up epochs, we optimize only LMean + LVar to obtain stable uncertainty estimates. After this warm-up phase, LAlign is activated.

3.6

95

Overall Objective Function

To jointly optimize the mean prediction, variance estimation, and feature manifold, the overall objective function is formulated as: L𝑇 𝑜𝑡𝑎𝑙 = L𝑀𝑒𝑎𝑛 + L𝑉 𝑎𝑟 + 𝜆L𝐴𝑙𝑖𝑔𝑛

(14)

(a) Vanilla

(b) FDS

where 𝜆 is a trade-off hyperparameter that balances the contribution of the alignment objective against the regression and variance estimation terms.

4

Experiment

We evaluate the proposed DUO on four deep imbalanced regression (DIR) datasets, spanning diverse tasks including age estimation, depth estimation, and protein mutation activity prediction. Specifically, AgeDB-DIR[24] and IMDB-WIKI-DIR[30] are utilized for large-scale continuous facial age regression. NYUD2-DIR, built upon NYU Depth V2[31], aims to predict depth maps from indoor RGB images. Furthermore, to validate the generalization capability of our method in bioinformatics, we introduce the AAV2-DIR dataset, which is constructed from high-throughput viability data of AAV2 capsid protein variants[6]. This comprehensive experimental setup covers typical scenarios ranging from 1D continuous targets to high-dimensional structured labels, designed to thoroughly verify the effectiveness and robustness of DUO in handling various extremely imbalanced data distributions.The label distributions of the four datasets are shown in the figure 3. Complete experimental results are provided in Tables S5–S8 of the Appendix.

4.1

Evaluation protocol and metrics

Following the standard DIR evaluation protocol, we report performance across four shot-based partitions: All (full test set), Many (>100 training samples per bin), Median (20–100 samples), and Few (<20 samples). For AgeDB-DIR, IMDB-WIKI-DIR and AAV2-DIR, we use MAE, balanced MAE (bMAE) [29], and error geometric mean (GM) [38], which measure overall error, bin-balanced error, and the geometric mean of per-sample errors, respectively. For NYUD2-DIR, we follow prior depth estimation DIR work and adopt RMSE and 𝑔 threshold accuracy 𝛿 1 (percentage of pixels with max( 𝑑𝑔 , 𝑑 ) < 1.25, 𝑔=ground-truth depth, 𝑑=predicted depth). 4.1.1 Main results for age estimation. Table 1 presents results on AgeDB-DIR and IMDB-WIKI-DIR, averaged over five random runs. DUO obtains the best few-shot bMAE and GM on both benchmarks. Its few-shot MAE remains competitive but is slightly higher than LDS on IMDB-WIKI-DIR (22.940 vs. 22.755) and Balanced MSE on AgeDB-DIR (9.917 vs. 9.690), revealing an empirical trade-off between tail-balanced performance and the best metric-specific error. This reflects an inherent head-tail trade-off in DIR, where minor head compromises (e.g., AgeDB Many-shot MAE 6.980 vs. Vanilla 6.743) are outweighed by substantial tail gains (Few-shot 9.917 vs. 13.381). 4.1.2 Main results for depth estimation. We further evaluate DUO on NYUD2-DIR to examine its effectiveness in structured

3 (c) RankSim

(d) DUO

Figure 4: Feature visualization on AgeDB-DIR for (a) VANILLA, (b) FDS, (c) Ranksim and (d) DUO.

regression with high-dimensional depth maps. Compared with existing DIR baselines, DUO achieves the best few-shot bMAE of 1.696, as shown in Table 2. These results indicate that DUO can be effectively extended from scalar regression to more complex dense prediction settings while preserving robust performance in tail regions. 4.1.3 Main results for protein activity estimation. Table 1 also summarizes AAV2-DIR, which maps protein sequences to functional fitness. DUO achieves the best few-shot MAE, bMAE, and GM of 4.106, 4.329, and 3.777, respectively. These results show that its few-shot advantages extend beyond visual regression to biological sequence activity prediction. 4.1.4 Feature visualizations. To examine representation differences on deep imbalanced regression (DIR) tasks, we use t-SNE[36] to project the penultimate-layer features of ResNet-50 into 2D and color samples by their continuous targets. As shown in Figure 4, Vanilla exhibits central collapse and blurred head–tail boundaries, FDS retains substantial local overlap, and RankSim forms an over-compressed line-like manifold. DUO instead produces a more continuous and less entangled layout. Since t-SNE is qualitative, Appendix Table S3 further reports KNN-MAE, where DUO obtains the lowest errors in the compound-tail and median-shot regions, quantitatively supporting improved local smoothness. 4.1.5 Pairwise comparison. Although our approach entails architectural modifications, we further evaluate its complementarity with existing methods. Table 4 compares AAV2 baselines with their DUO-enhanced counterparts, with the better value in each pair highlighted in bold. DUO integration improves MAE and bMAE across all pairs and improves most GM values, although RankSim’s GM slightly worsens. This indicates useful but metric-dependent

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression

Table 1: Results are presented for the few-shot region on the IMDB, AgeDB, and AAV2 datasets. The first section reports the results of the baselines and our method, with the best results highlighted in bold. The second section reports the performance differences with respect to the corresponding baselines MAE

bMAE

GM

IMDB-DIR AgeDB-DIR AAV2-DIR IMDB-DIR AgeDB-DIR AAV2-DIR IMDB-DIR AgeDB-DIR AAV2-DIR Vanilla LDS FDS Ranksim ConR Balanced MSE Distloss DUO (Ours)

25.546 22.755 24.347 25.317 25.969 23.541 24.215 22.940

13.381 11.489 12.136 14.601 12.333 9.690 12.121 9.917

7.365 4.361 5.023 4.885 4.183 7.283 7.432 4.106

33.085 30.179 31.839 32.464 33.601 30.000 30.165 27.190

16.579 12.055 15.531 17.611 14.486 11.940 13.670 11.590

7.565 4.681 5.388 5.225 4.335 7.563 7.567 4.329

18.091 13.786 14.600 17.639 17.990 13.803 14.076 13.650

9.644 7.577 8.326 11.021 9.376 10.480 7.726 6.587

7.351 4.054 4.786 4.652 3.932 7.253 7.416 3.777

Ours vs. Vanilla Ours vs. LDS Ours vs. FDS Ours vs. Ranksim Ours vs. ConR Ours vs. BM Ours vs. Distloss

+ 2.606 - 0.185 + 1.407 + 2.377 + 3.029 + 0.601 + 1.275

+ 3.464 + 1.572 + 2.219 + 4.684 + 2.416 - 0.227 + 2.204

+ 3.259 + 0.255 + 0.917 + 0.779 + 0.077 + 3.177 + 3.326

+ 5.895 + 2.989 + 4.649 + 5.274 + 6.411 + 2.810 + 2.975

+ 4.989 + 0.465 + 3.941 + 6.021 + 2.896 + 0.350 + 2.080

+ 3.236 + 0.352 + 1.059 + 0.896 + 0.006 + 3.234 + 3.238

+ 4.441 + 0.136 + 0.950 + 3.989 + 4.340 + 0.153 + 0.426

+ 3.057 + 0.990 + 1.739 + 4.434 + 2.789 + 3.893 + 1.139

+ 3.574 + 0.277 + 1.009 + 0.875 + 0.155 + 3.476 + 3.639

Table 2: Experimental results on the few-shot region of the NYUD2 dataset. Note that we do not report results for twostage training methods like Balanced MSE and DistLoss on NYUD2, as these methods are primarily designed for global regression and do not directly scale to structured pixel-wise depth prediction. Method

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

Vanilla LDS FDS RankSim ConR DUO(Ours)

1.867 1.764 1.905 1.896 1.869 1.759

1.443 1.333 1.478 1.475 1.459 1.351

1.884 1.714 1.920 1.907 1.912 1.696

0.608 0.641 0.594 0.601 0.591 0.643

Ours vs. Vanilla Ours vs. LDS Ours vs. FDS Ours vs. RankSim Ours vs. ConR

+ 0.108 + 0.005 + 0.146 + 0.137 + 0.110

+ 0.092 - 0.018 + 0.127 + 0.124 + 0.108

+ 0.188 + 0.018 + 0.224 + 0.211 + 0.216

+ 0.035 + 0.002 + 0.049 + 0.042 + 0.052

complementarity. Results on the other datasets are provided in Appendix Tables S9–S11.

4.2

Ablation Studies on Design Modules

4.2.1 Component-wise ablation of DUO. Table 3 shows that the decoupled objective already improves few-shot performance, while the full DUO further benefits from contrastive alignment. To isolate capacity, decoupling, uncertainty weighting, and BCbased assignment, Appendix Table S1 reports controlled variants under matched dual-head capacity. The results show that capacity

(a) Visualization for 𝝀 = 𝟎. 𝟓

(b) Visualization for 𝝀 = 𝟏. 𝟎

𝑀𝐴𝐸

𝝀 (c) Analysis of the hyper-parameter𝝀

Figure 5: Ablation study on the hyper-parameter 𝜆. (a) and (b) compares the learnt feature space for parameter of 0.5 and 1 respectively. (c) shows the MAE performance under various 𝜆 values. alone is insufficient and that removing weighting or replacing BC consistently degrades few-shot metrics. 4.2.2 Analysis of the hyper-parameter 𝜆. We further analyze the impact of hyper-parameter 𝜆, which controls the weight of DUO’s contrastive term and governs the trade-off between samplelevel fitting and structure-level constraints during optimization.

Juncheng Zhou, Jiaxi Lu, Weijing Zeng, Zhong Li, Hao Qi, & Jingsong Cui

Table 3: Component-wise ablation study of DUO in the few-shot regions. Vanilla denotes the baseline trained with Mean Squared Error (MSE). DUO-C removes the contrastive term, and DUO uses the complete objective. The best results are highlighted in bold. MAE ↓

bMAE ↓

GM ↓

Method

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

Vanilla DUO-C DUO

25.546 23.332 22.940

13.381 10.787 9.917

7.365 4.193 4.106

33.085 30.349 27.190

16.579 13.092 11.590

7.565 4.400 4.329

18.091 13.862 13.650

9.644 6.808 6.587

7.351 3.948 3.777

Table 4: Pairwise comparison of various baselines and their DUO-enhanced counterparts on the AAV2 dataset across different shot regions. The best results within each comparison pair are highlighted in bold. Overall

Few

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

LDS LDS+DUO

1.990 1.721

1.981 1.718

1.178 1.060

4.361 4.159

4.681 4.329

4.054 3.770

3.648 3.253

3.888 3.440

3.211 2.800

1.935 1.669

2.117 1.817

1.140 1.027

FDS FDS+DUO

1.988 1.794

1.985 1.791

1.256 1.128

5.023 4.379

5.388 4.542

4.786 3.984

4.345 3.603

4.667 3.854

4.008 3.273

1.911 1.734

2.147 1.923

1.210 1.090

RankSim RankSim+DUO

1.936 1.831

1.932 1.829

1.189 1.231

4.885 4.796

5.225 5.195

4.652 4.408

4.151 4.002

4.408 4.279

3.857 3.651

1.864 1.759

2.110 1.977

1.145 1.189

Balanced MSE (GAI) Balanced MSE+DUO

2.650 2.439

2.639 2.441

2.115 1.839

7.283 6.677

7.563 7.076

7.253 6.576

5.992 5.693

6.226 5.968

5.978 5.598

2.539 2.333

2.745 2.573

2.045 1.775

As shown in Figure 5, a small 𝜆 imposes insufficient contrastive constraints, resulting in training dynamics close to standard regression and limited performance gains in the few-shot region. With a moderate increase in 𝜆, few-shot region performance improves consistently, as stronger structural constraints effectively mitigate the representation shift induced by imbalanced data distributions. However, an excessively large 𝜆 degrades overall performance across all regions, as overly strong auxiliary constraints disrupt the primary regression objective and hinder the model from balancing performance across target intervals. Feature visualizations further corroborate these trends: a moderate 𝜆 yields a clearer, more continuous feature structure with well-defined inter-region boundaries and minimal overlap between minority and majority samples, while an overly large 𝜆 causes feature space over-separation or local instability. Based on both quantitative results and visual analysis, we fix 𝜆 = 1.0 for all subsequent experiments.

5

Median

Conclusion

We presented DUO, an uncertainty-aware framework for deep imbalanced regression. By modeling each target as a conditional Gaussian distribution, DUO computationally decouples mean and variance optimization and refines feature learning through distributionguided contrastive alignment. Experiments on AgeDB-DIR, IMDBWIKI-DIR, NYUD2-DIR, and AAV2-DIR show the strongest gains in scarce regions, particularly on imbalance-aware metrics, with modest trade-offs on some overall or many-shot results. These findings support sample-wise uncertainty as an effective signal for robust long-tailed regression.

Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression

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Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” JUNCHENG ZHOU∗ , School of Cyber Science and Engineering, Wuhan University, China JIAXI LU∗ , School of Cyber Science and Engineering, Wuhan University, China WEIJING ZENG, School of Mathematics and Statistics, Wuhan University, China ZHONG LI, School of Synthetic Biology and Biomanufacturing, Tianjin University, China HAO QI, School of Synthetic Biology and Biomanufacturing, Tianjin University, China JINGSONG CUI✉ , School of Cyber Science and Engineering, Wuhan University, China A

Proof of Maximum Entropy Optimality for Gaussian Distribution

In this section, we demonstrate from an information-theoretic perspective that the Gaussian distribution is the unique optimal probabilistic form that introduces the minimum inductive bias under given mean 𝜇 and variance 𝜎 2 constraints, achieved by maximizing the differential entropy[4]. Theorem A.1 (Maximum Entropy Distribution). Let 𝑋 be a continuous random variable defined on (−∞, +∞) with a probability density function 𝑝 (𝑥). Among all possible distributions satisfying E[𝑋 ] = 𝜇 and Var(𝑋 ) = 𝜎 2 , the unique distribution that maximizes the differential entropy 𝐻 (𝑝) is the Gaussian distribution N (𝜇, 𝜎 2 ). ∫ +∞ Proof. The differential entropy of a continuous random variable 𝑋 is defined as 𝐻 (𝑝) = − −∞ 𝑝 (𝑥) ln 𝑝 (𝑥) 𝑑𝑥. To find the distribution 𝑝 (𝑥) that maximizes 𝐻 (𝑝), we introduce the following constraints: ∫ +∞ • Normalization: −∞ 𝑝 (𝑥) 𝑑𝑥 = 1 ∫ +∞ • Mean: −∞ 𝑥𝑝 (𝑥) 𝑑𝑥 = 𝜇 ∫ +∞ • Variance: −∞ (𝑥 − 𝜇) 2 𝑝 (𝑥) 𝑑𝑥 = 𝜎 2 We construct the Lagrangian functional L (𝑝) incorporating these constraints: ∫ +∞  ∫ +∞ L (𝑝) = − 𝑝 (𝑦) ln 𝑝 (𝑦) 𝑑𝑦 + 𝜆0 𝑝 (𝑦) 𝑑𝑦 − 1 −∞

−∞

∫ +∞ + 𝜆1

−∞

 𝑦𝑝 (𝑦) 𝑑𝑦 − 𝜇 + 𝜆2

∫ +∞

(𝑦 − 𝜇) 2 𝑝 (𝑦) 𝑑𝑦 − 𝜎 2



−∞

Setting the functional derivative with respect to 𝑝 (𝑦) to zero yields the extremum condition: 𝛿L = −1 − ln 𝑝 (𝑦) + 𝜆0 + 𝜆1𝑦 + 𝜆2 (𝑦 − 𝜇) 2 = 0 𝛿𝑝 (𝑦) ∗ Juncheng Zhou and Jiaxi Lu contributed equally to this work.

Accepted at the 34th ACM International Conference on Multimedia (ACM MM 2026). The Version of Record will be available at https://doi.org/10.1145/ 3767308.3835898. Authors’ Contact Information: Juncheng Zhou, School of Cyber Science and Engineering, Wuhan University, Wuhan, China, [email protected]; Jiaxi Lu, School of Cyber Science and Engineering, Wuhan University, Wuhan, China, [email protected]; Weijing Zeng, School of Mathematics and Statistics, Wuhan University, Wuhan, China, [email protected]; Zhong Li, School of Synthetic Biology and Biomanufacturing, Tianjin University, Tianjin, China, [email protected]; Hao Qi, School of Synthetic Biology and Biomanufacturing, Tianjin University, Tianjin, China, [email protected]; Jingsong Cui, School of Cyber Science and Engineering, Wuhan University, Wuhan, China, [email protected].

1

2

Zhou et al.

which implies 𝑝 (𝑦) = exp (𝜆0 − 1) + 𝜆1𝑦 + 𝜆2 (𝑦 − 𝜇) 2 . For integrability we require 𝜆2 < 0. And for a Gaussian-type 2

distribution 𝑝 (𝑦) ∝ 𝑒 𝑎𝑦 +𝑏𝑦 with 𝑎 < 0, the mean is E[𝑦] = −

𝑏 . 2𝑎

Substituting 𝑎 = 𝜆2 and 𝑏 = 𝜆1 − 2𝜆2 𝜇 yields E[𝑦] = −

𝜆1 − 2𝜆2 𝜇 𝜆1 =𝜇− . 2𝜆2 2𝜆2

Using the mean constraint E[𝑦] = 𝜇, we have 𝜇− Since 𝜆2 < 0, we know 2𝜆2 ≠ 0, which implies

𝜆1 = 𝜇. 2𝜆2

𝜆1 = 0. Let 𝐶 = exp(𝜆0 − 1). Substituting 𝑝 (𝑦) into the constraints and utilizing Gaussian integrals: √︂ √ ∫ +∞ ∫ +∞ 𝐶 𝜋 2 2 𝜋 , 𝜎2 = (𝑦 − 𝜇) 2𝐶𝑒 𝜆2 (𝑦−𝜇 ) 𝑑𝑦 = 1= 𝐶𝑒 𝜆2 (𝑦−𝜇 ) 𝑑𝑦 = 𝐶 −𝜆2 2(−𝜆2 ) 3/2 −∞ −∞ √ 3/2 𝐶 𝜋/2(−𝜆2 ) 1 1 1 =⇒ 𝜎 2 = =⇒ 𝜆2 = − 2 , 𝐶 = √ = √︁ −2𝜆 2𝜎 2 2𝜋𝜎 2 𝐶 𝜋/−𝜆2 Substituting 𝜆2 and 𝐶 back into the probability density function, we obtain:   (𝑦 − 𝜇) 2 1 exp − 𝑝 (𝑦) = √ 2𝜎 2 2𝜋𝜎 2

(1)

The derivation rigorously establishes the optimal probabilistic form for instance-level variance modeling from an information-theoretic perspective. B

□

Optimization Laziness Analysis of NLL Loss in Deep Imbalanced Regression

B.1

Problem Formulation and Notation

Consider a heteroscedastic Gaussian regression model where for an input 𝑥, the model outputs mean 𝜇 (𝑥) and variance 𝜎 2 (𝑥) > 0. Given label 𝑦, the single-sample Negative Log-Likelihood (NLL) loss is [7]: L𝑁 𝐿𝐿 =

(𝑦 − 𝜇 (𝑥)) 2 1 log 𝜎 2 (𝑥) + 2 2𝜎 2 (𝑥)

(2)

For simplicity, we define the residual 𝑒 (𝑥) := 𝑦 − 𝜇 (𝑥) and let 𝑣 := 𝜎 2 (𝑥) > 0 be the optimization variable. The loss 2

becomes L = 12 log 𝑣 + 𝑒2𝑣 . In this analysis, we study partial derivatives with respect to 𝜇 and 𝑣 separately; hence, when differentiating with respect to 𝑣, the residual 𝑒 = 𝑦 − 𝜇 is treated as fixed. B.2

Gradient Computation

Proposition B.1. The first-order partial derivatives of L with respect to 𝜇 and 𝑣 = 𝜎 2 are: ∇𝜇 L =

𝜇 −𝑦 𝑒 = − 2, 𝜎 2 (𝑥) 𝜎

∇𝑣 L =

𝜎 2 (𝑥) − 𝑒 2 2𝜎 4 (𝑥)

Proof. Fixing 𝑣, the derivative w.r.t. 𝜇 is:   2(𝑦 − 𝜇)(−1) 𝜇 − 𝑦 𝜕L 𝜕 (𝑦 − 𝜇) 2 𝑒 = = = =− 2 𝜕𝜇 𝜕𝜇 2𝑣 2𝑣 𝑣 𝜎

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Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 3 𝜕𝑒 where 𝜕𝜇 = −1. The mean gradient is scaled by 1/𝜎 2 (𝑥); thus, a large 𝜎 2 (𝑥) directly suppresses the update step of 𝜇.

Fixing 𝜇, the derivative w.r.t. 𝑣 is:   𝜕L 𝜕 1 𝑒2 1 𝑒2 𝑣 − 𝑒 2 𝜎 2 (𝑥) − 𝑒 2 = log 𝑣 + = − 2 = = 𝜕𝑣 𝜕𝑣 2 2𝑣 2𝑣 2𝑣 2𝑣 2 2𝜎 4 (𝑥) □ B.3

Convergence of Standard Iterative Methods to a Unique Optimum

B.3.1

Gradient Descent (GD).

Theorem B.2 (Variance Inflation and Convergence). Under the GD update rule with learning rate 𝜂 > 0, if 𝑒 2 > 𝑣𝑡 , the variance strictly increases. The unique stable equilibrium is 𝑣 ∗ = 𝑒 2 , around which GD converges linearly. 2

Proof. (a) Monotonicity: The GD rule is 𝑣𝑡 +1 = 𝑣𝑡 −𝜂 𝜕𝜕𝑣L 𝑣𝑡 = 𝑣𝑡 −𝜂 𝑣𝑡2𝑣−𝑒2 . If 𝑒 2 > 𝑣𝑡 , then 𝜕𝜕𝑣L < 0, leading to 𝑣𝑡 +1 > 𝑣𝑡 . 𝑡

Thus, the variance strictly increases.

(b) Uniqueness, Lipschitz Smoothness and Step Size Condition: Setting 𝜕𝜕𝑣L = 0 yields the unique critical point 𝑣 ∗ = 𝑒 2 . The second-order derivative (Hessian) is   𝜕2 L 𝜕 1 𝑒2 1 𝑒 2 2𝑒 2 − 𝑣 = − =− 2 + 3 = . 2 2 𝜕𝑣 𝜕𝑣 2𝑣 2𝑣 2𝑣 𝑣 2𝑣 3 In a neighborhood of 𝑣 ∗ = 𝑒 2 , the Hessian is positive and bounded above by 𝐿 = 2𝑒14 , which implies the gradient of L is

𝐿-Lipschitz continuous in this neighborhood. For gradient descent to converge stably to a local minimum, the step size 2

must satisfy 0 < 𝜂 < 𝐿2 = 4𝑒 4 . At 𝑣 = 𝑒 2 , we have 𝜕𝜕𝑣L2

𝑣=𝑒 2

= 2𝑒14 > 0, so 𝑣 ∗ = 𝑒 2 is a strict local minimum.

(c) Linear Convergence Rate: Let 𝛿𝑡 = 𝑣𝑡 − 𝑒 2 . A first-order Taylor expansion near 𝑣 ∗ gives 𝛿𝑡 +1 ≈ (1 − 𝑒 4 )𝛿𝑡 + O (𝛿𝑡2 ). 𝜂

With 0 < 𝜂 < 2𝑒 4 , the convergence factor 𝜌 = |1 − 𝜂/𝑒 4 | < 1 ensures linear convergence: 𝛿𝑡 = O (𝜌 𝑡 ).

□

In summary, under the approximate equilibrium state 𝜎 2 (𝑥) ≈ 𝑒 (𝑥) 2 , the predicted variance of "hard" samples (where 𝑒 2 ≫ 𝜎 2 ) will rapidly inflate to the magnitude of the squared residual. B.3.2

Newton’s Method.

Theorem B.3 (Quadratic Convergence under Newton’s Method). When updating the variance using Newton’s method, the iteration satisfies:

𝑣𝑡 (3𝑒 2 − 2𝑣𝑡 ) 2𝑒 2 − 𝑣𝑡 Furthermore, the deviation 𝛿𝑡 = 𝑣𝑡 − 𝑒 2 achieves quadratic convergence: 𝑣𝑡 +1 =

|𝛿𝑡 +1 | =

2𝛿𝑡2 2 𝑒 −𝛿

= O (𝛿𝑡2 )

(4)

(5)

𝑡

Proof. (a) Derivation of Newton’s update formula: The update rule is 𝑣𝑡 +1 = 𝑣𝑡 − [∇𝑣2 L] −1 ∇𝑣 L. Substituting the 2

2

known derivatives ∇𝑣 L = 𝑣−𝑒 and ∇𝑣2 L = 2𝑒2𝑣−𝑣 3 , the iteration step size Δ𝑣 and the updated variance 𝑣 𝑡 +1 are computed 2𝑣 2 as:

2𝑣 3 𝑣 − 𝑒 2 𝑣 (𝑒 2 − 𝑣) · = 2 2𝑒 − 𝑣 2𝑣 2 2𝑒 2 − 𝑣 2 2 𝑣 (𝑒 − 𝑣) 𝑣 (2𝑒 − 𝑣) + 𝑣 (𝑒 2 − 𝑣) 𝑣 (3𝑒 2 − 2𝑣) 𝑣𝑡 +1 = 𝑣 + = = 2𝑒 2 − 𝑣 2𝑒 2 − 𝑣 2𝑒 2 − 𝑣 Δ𝑣 = −

4

Zhou et al. 2

2

2

−2𝑒 ) (b) Verification of the equilibrium point: Letting 𝑣 = 𝑒 2 , we obtain 𝑣𝑡 +1 = 𝑒 (3𝑒 = 𝑒2. 2𝑒 2 −𝑒 2

(c) Quadratic convergence rate: Newton’s method converges quadratically to 𝑣 ∗ for initial points sufficiently close to 𝑣 ∗ , since the Hessian ∇𝑣2 L (𝑣 ∗ ) = 2𝑒14 > 0 is positive definite and non-singular at 𝑣 ∗ . Let 𝛿𝑡 = 𝑣𝑡 − 𝑒 2 . Substituting

𝑣𝑡 = 𝑒 2 + 𝛿𝑡 into the update formula, the deviation at the next step 𝛿𝑡 +1 = 𝑣𝑡 +1 − 𝑒 2 is: 𝛿𝑡 +1 =

𝑒 4 − 𝑒 2𝛿𝑡 − 2𝛿𝑡2 − 𝑒 4 + 𝑒 2𝛿𝑡 −2𝛿𝑡2 (𝑒 2 + 𝛿𝑡 )(𝑒 2 − 2𝛿𝑡 ) 2 − 𝑒 = = 𝑒 2 − 𝛿𝑡 𝑒 2 − 𝛿𝑡 𝑒 2 − 𝛿𝑡 2𝛿 2

For sufficiently small 𝛿𝑡 , 𝑒 2 − 𝛿𝑡 > 0, ensuring 𝑣𝑡 > 0 throughout the iteration. Thus, |𝛿𝑡 +1 | = |𝑒 2 −𝛿𝑡 | = O (𝛿𝑡2 ). This 𝑡

demonstrates that the residual converges quadratically. B.4

□

Proof of the Vanishing Mean Update Signal

Theorem B.4 (Gradient Vanishing). Under the approximate equilibrium state 𝜎 2 (𝑥) ≈ 𝑒 (𝑥) 2 : ∇𝜇 L =

|𝑒 | |𝑒 | 1 ≈ 2 = 𝜎2 𝑒 |𝑒 |

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Consequently, lim |𝑒 (𝑥 ) |→∞ ∇𝜇 L = 0. That is, the mean update signal for hard samples (with extremely large errors) tends to vanish. Proof. we have ∇𝜇 L = − 𝜎𝑒2 , thus the gradient magnitude is: ∥∇𝜇 L ∥ =

|𝑒 | . 𝜎2 2

2

= 0. Substituting into the By the equilibrium condition, 𝜎 2 = 𝑒 2 + 𝑜 (𝑒 2 ) as |𝑒 | → ∞, which means lim |𝑒 |→∞ 𝜎 𝑒−𝑒 2 gradient:

|𝑒 | 1 1 |𝑒 | = = · . 𝜎 2 𝑒 2 + 𝑜 (𝑒 2 ) |𝑒 | 1 + 𝑜 (𝑒 2 ) 𝑒2

2) As |𝑒 | → ∞, 𝑜 (𝑒 → 0, so 𝑜1(𝑒 2 ) 𝑒2 1+

→ 1, hence:

𝑒2

|𝑒 | 1 ∼ 𝜎2 |𝑒 |

(|𝑒 | → ∞).

Taking the limit: lim ∥∇𝜇 L ∥ = lim

|𝑒 |→∞

1

|𝑒 |→∞ |𝑒 |

= 0.

This completes the proof that the mean update signal vanishes for hard samples.

□

□

To rapidly decrease the NLL loss for hard samples, the optimizer inflates the predicted variance. However, this simultaneously suppresses the learning signal for the mean to zero, rendering the model almost unable to improve its mean predictions for these challenging samples. Whether using Gradient Descent or Newton’s method, the optimization of the NLL loss drives 𝜎 2 (𝑥) → 𝑒 (𝑥) 2 =
2 𝑦 − 𝜇 (𝑥) . Since Gradient Descent converges linearly and Newton’s method converges quadratically, the adaptive variance actively “absorbs” the errors of hard samples during training. As a result, the mean head nearly stops learning for these instances.

Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 5 Convergence Measure Proof for L𝑉 𝑎𝑟 Loss

C

Proof Objective: For nonzero residuals, we identify the unique global minimum of L𝑉 𝑎𝑟 and characterize its local optimization behavior in the linear and logarithmic parameterizations. C.1

Problem Formulation

C.1.1

Loss Function. The variance-specific loss L𝑉 𝑎𝑟 is defined as:   (𝑦 − sg[ 𝜇ˆ (𝑥)]) 2 1 2 ˆ L𝑉 𝑎𝑟 = 𝛽 · + log 𝜎 (𝑥) 2𝜎ˆ 2 (𝑥) 2

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where: • sg[·] denotes the stop-gradient operator, which treats the mean prediction 𝜇ˆ (𝑥) as a constant during variance optimization to decouple the two processes. • 𝛽 > 0 is a positive scaling coefficient. • 𝜎ˆ 2 (𝑥) > 0 is the predicted variance to be optimized. C.1.2 Notation Simplification. For a fixed sample (𝑥, 𝑦) with a nonzero residual, let 𝐸 := (𝑦 − sg[ 𝜇ˆ (𝑥)]) 2 > 0 and 𝑆 := 𝜎ˆ 2 (𝑥) > 0. The objective function simplifies to a univariate function of 𝑆:   𝐸 1 + log 𝑆 , 𝑆 > 0 𝐽 (𝑆) = 𝛽 2𝑆 2

(8)

Under logarithmic parameterization (let 𝑣 = log 𝜎ˆ 2 (𝑥), hence 𝑆 = 𝑒 𝑣 ): 𝛽 𝐽˜ (𝑣) = (𝐸𝑒 −𝑣 + 𝑣) , 2 C.2

Convergence Analysis in Linear Space

C.2.1

Global Unique Minimum.

𝑣∈R

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Theorem C.1. The function 𝐽 (𝑆) has a unique global minimum on (0, +∞) at: 𝑆 ∗ = 𝐸 = (𝑦 − 𝜇ˆ (𝑥)) 2

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𝛽 (𝑆 −𝐸 ) . It is negative for 0 < 𝑆 < 𝐸, zero only at 𝑆 = 𝐸, and positive for 𝑆 > 𝐸. 2𝑆 2 𝛽 (2𝐸−𝑆 ) Hence, 𝐽 decreases up to 𝐸 and increases thereafter, so 𝑆 ∗ = 𝐸 is the unique global minimum. Notice that 𝐽 ′′ (𝑆) = 2𝑆 3

Proof. The derivative is 𝜕𝐽𝜕𝑆(𝑆 ) =

changes sign; global convexity is therefore neither claimed nor required. Thus, L𝑉 𝑎𝑟 and L𝑁 𝐿𝐿 share the same stationary variance for a fixed mean prediction.

□

C.2.2 Convergence Rate in Linear Space. A critical distinction lies in the gradient dynamics. For L𝑁 𝐿𝐿 , the mean 𝑁 𝐿𝐿 gradient 𝜕 L𝜕𝜇 = − 𝜎 2 (𝑥 ) is suppressed by a factor of 𝑐 when 𝜎 2 (𝑥) inflates (𝜎 2 = 𝑐𝐸, 𝑐 ≫ 1). In contrast, for L𝑉 𝑎𝑟 ,

𝑦−𝜇 (𝑥 )

𝑉 𝑎𝑟 the stop-gradient operator ensures 𝜕 L𝜕𝜇 ≡ 0. The mean head is updated via an independent loss term, making 𝑉 𝑎𝑟 term

its gradient magnitude invariant to 𝜎 2 inflation.

Comparing the Hessians at 𝑆 = 𝐸: 𝐻𝑉 𝑎𝑟 = 2𝐸 2 = 𝛽 · 𝐻 𝑁 𝐿𝐿 . Under Gradient Descent (GD) with step size 𝜂, the spectral 𝛽

radii are:

𝜂 𝜂𝛽 |, 𝜌𝑉 𝑎𝑟 = |1 − 2 | (11) 2𝐸 2 2𝐸 The coefficient 𝛽 therefore rescales the local curvature and must be considered jointly with the optimization step size. 𝜌 𝑁 𝐿𝐿 = |1 −

6

Zhou et al. Theorem C.2 (Quadratic Convergence of Newton’s Method). Let 𝛿𝑡 = 𝑆𝑡 − 𝐸 be the deviation at step 𝑡. When

initialized sufficiently close to 𝐸, Newton’s method on 𝐽 (𝑆) achieves quadratic convergence: |𝛿𝑡 +1 | = O (𝛿𝑡2 ). ′

𝑆𝑡 (3𝐸−2𝑆𝑡 ) 𝑡) Proof. The Newton update 𝑆𝑡 +1 = 𝑆𝑡 − 𝐽𝐽 ′′(𝑆 . Substituting 𝑆𝑡 = 𝐸 + 𝛿𝑡 : 2𝐸−𝑆𝑡 (𝑆𝑡 ) simplifies to 𝑆𝑡 +1 =

𝛿𝑡 +1 =

𝐸 2 − 𝐸𝛿𝑡 − 2𝛿𝑡2 − 𝐸 2 + 𝐸𝛿𝑡 −2𝛿𝑡2 (𝐸 + 𝛿𝑡 )(𝐸 − 2𝛿𝑡 ) −𝐸 = = 𝐸 − 𝛿𝑡 𝐸 − 𝛿𝑡 𝐸 − 𝛿𝑡

2𝛿 2

Thus, |𝛿𝑡 +1 | = |𝐸−𝛿𝑡 𝑡 | = O (𝛿𝑡2 ). C.2.3

□

Convergence under 𝜒 2 -Divergence.

Theorem C.3. Locally, GD optimization of 𝐽 (𝑆) induces a contraction in the 𝜒 2 -divergence space. Defining 𝐷 𝜒 2 (𝑆 ∥𝐸) :=

(𝑆 −𝐸 ) 2 , the iteration satisfies: 𝐸

𝐷 𝜒 2 (𝑆𝑡 +1 ∥𝐸) = (1 −

𝜂𝛽 2 ) 𝐷 𝜒 2 (𝑆𝑡 ∥𝐸) + O (|𝛿𝑡 | 3 ) 2𝐸 2

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Proof. Expanding 𝛿𝑡 +1 = 𝛿𝑡 − 𝜂 2𝑆 2𝑡 near 𝑆𝑡 ≈ 𝐸, we get 𝛿𝑡 +1 = 𝜌𝛿𝑡 + O (𝛿𝑡2 ) where 𝜌 := 1 − 2𝐸 2 . For |𝜌 | < 1, squaring 𝛽𝛿

𝜂𝛽

𝑡

this local expansion gives 𝐷 𝜒 2 (𝑆𝑡 +1 ∥𝐸) = 𝜌 2 𝐷 𝜒 2 (𝑆𝑡 ∥𝐸) + O (|𝛿𝑡 | 3 ). D

□

Closed-form Derivation of the Bhattacharyya Coefficient for Gaussian Distributions

This appendix provides the detailed mathematical derivation for the closed-form Bhattacharyya Coefficient (BC) between two label-anchored Gaussian proxy distributions, 𝑝 (𝑦|𝑥𝑖 ) = N (𝑦𝑖 , 𝜎ˆ𝑖2 ) and 𝑝 (𝑦|𝑥 𝑗 ) = N (𝑦 𝑗 , 𝜎ˆ 𝑗2 ), as presented in Section 3.5.1. By substituting the Gaussian density functions into the overlap integral and completing the square for the exponential terms, we have: ∫ +∞ √︁ 𝑝 (𝑦|𝑥𝑖 )𝑝 (𝑦|𝑥 𝑗 ) 𝑑𝑦 −∞ v ! u   ∫ +∞ u t 1 (𝑦 − 𝑦 𝑗 ) 2 (𝑦 − 𝑦𝑖 ) 2 1 · √︃ = exp − exp − 𝑑𝑦 √︃ 2𝜎ˆ𝑖2 2𝜎ˆ 𝑗2 −∞ 2𝜋 𝜎ˆ 2 2𝜋 𝜎ˆ 2

BC(𝑖, 𝑗) =

𝑗

𝑖

1 = (2𝜋 𝜎ˆ𝑖 𝜎ˆ 𝑗 ) 1/2

∫ +∞ −∞

1 exp − 4

"

(𝑦 − 𝑦𝑖 ) 2 (𝑦 − 𝑦 𝑗 ) 2 + 𝜎ˆ𝑖2 𝜎ˆ 𝑗2

#! 𝑑𝑦

!2  2  2 𝑦𝑖 𝜎ˆ 𝑗2 + 𝑦 𝑗 𝜎ˆ𝑖2 (𝑦𝑖 − 𝑦 𝑗 ) 2  ª © 1  𝜎ˆ𝑖 + 𝜎ˆ 𝑗 = √︁ exp ­−  2 2 𝑦 − + ® 𝑑𝑦 4  𝜎ˆ𝑖 𝜎ˆ 𝑗 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2  2𝜋 𝜎ˆ𝑖 𝜎ˆ 𝑗 −∞ «  ¬ !∫ ! 2 + 𝜎ˆ 2 2 + 𝑦 𝜎ˆ 2 2 2 +∞ ˆ ˆ 𝜎 𝑦 𝜎 𝑖 𝑗 𝑗 𝑖 (𝑦𝑖 − 𝑦 𝑗 ) 1 𝑗 © 𝑖 ª exp − = √︁ exp − 𝑦 − ­ ® 𝑑𝑦 4(𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 ) −∞ 4𝜎ˆ𝑖2 𝜎ˆ 𝑗2 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 2𝜋 𝜎ˆ𝑖 𝜎ˆ 𝑗 « ¬ t ! v ! √︄ 2 𝜎ˆ 2 2 2 ˆ 4𝜋 𝜎 (𝑦𝑖 − 𝑦 𝑗 ) (𝑦𝑖 − 𝑦 𝑗 ) 2𝜎ˆ𝑖 𝜎ˆ 𝑗 1 𝑖 𝑗 exp − = √︁ · = exp − 4(𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 ) 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 4(𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 ) 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 2𝜋 𝜎ˆ𝑖 𝜎ˆ 𝑗 ! 2 2 1 (𝑦𝑖 − 𝑦 𝑗 ) 2 1 𝜎ˆ𝑖 + 𝜎ˆ 𝑗 − ln = exp − 4 𝜎ˆ𝑖2 + 𝜎ˆ 𝑗2 2 2𝜎ˆ𝑖 𝜎ˆ 𝑗 1

∫ +∞

This completes the derivation, mathematically verifying the closed-form formulation of our adaptive semantic tolerance measure.

Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 7 E

Asymptotic Analysis of Repulsive Gradients Based on Reciprocal Distribution Overlap

This section mathematically analyzes the proposed weighting mechanism based on reciprocal distribution overlap. We demonstrate that when sample pairs are in a state of Topological Hallucination, this mechanism adaptively generates exponentially amplified repulsive gradients, effectively correcting erroneous connections within the feature manifold. Proposition E.1 (Asymptotic Growth Rate of Repulsive Gradients). Assume samples 𝑖 and 𝑘 are in a state of topological hallucination, where their semantic label distance is large (Δ𝑖,𝑘 = |𝑦𝑖 − 𝑦𝑘 | → ∞) despite being erroneously clustered in the feature space (𝑐𝑖,𝑘 → 1). Suppose the predicted variances are bounded, i.e., 𝜎ˆ𝑖 , 𝜎ˆ𝑘 = O (1). For any ordinary hard negative sample 𝑚 with a finite label distance, the relative repulsive gradient produced by target sample 𝑘 asymptotically satisfies:

2 Δ𝑖,𝑘 𝐹 (𝑖, 𝑘) ∼ exp 𝐹 (𝑖, 𝑚) Σ2

! (13)

→ +∞

where Σ2 = 4(𝜎ˆ𝑖2 + 𝜎ˆ𝑘2 ). That is, the repulsive gradient grows squared-exponentially with respect to the label distance. Proof. First, we derive the explicit expression for the repulsive gradient. Let 𝑐𝑖,𝑘 = cos(𝑧𝑖 , 𝑧𝑘 ) denote the cosine similarity between samples 𝑖 and 𝑘. Based on the definition of the alignment loss L𝐴𝑙𝑖𝑔𝑛 , the partial derivative with respect to 𝑐𝑖,𝑘 for a negative sample 𝑘 ∈ N𝑖 yields the base repulsive gradient: (𝑖 ) 𝜕L𝐴𝑙𝑖𝑔𝑛

1 𝑐𝑖,𝑘 /𝜏 = 𝑒 𝜕𝑐𝑖,𝑘 𝜏𝑍𝑖 Í where 𝜏 is the temperature parameter and 𝑍𝑖 = 𝑗 𝑒 𝑐𝑖,𝑗 /𝜏 is the partition function serving as the normalization denominator. Incorporating our proposed dynamic weight W𝑝𝑢𝑠ℎ (𝑖, 𝑘) = 𝑤 (𝑦𝑖 ) · BC(𝑖, 𝑘) −1 , the full repulsive gradient 𝐹 (𝑖, 𝑘) becomes:

𝑤 (𝑦𝑖 ) · BC(𝑖, 𝑘) −1 · 𝑒 𝑐𝑖,𝑘 /𝜏 (14) 𝜏𝑍𝑖 Next, we expand the reciprocal Bhattacharyya weight. For the Gaussian proxies 𝑝 (𝑦|𝑥𝑖 ) = N (𝑦𝑖 , 𝜎ˆ𝑖2 ), the closed-form 𝐹 (𝑖, 𝑘) =

reciprocal BC is: BC(𝑖, 𝑘)

−1

2 Δ𝑖,𝑘

2 2 1 𝜎ˆ𝑖 + 𝜎ˆ𝑘 + ln = exp 2𝜎ˆ𝑖 𝜎ˆ𝑘 4(𝜎ˆ𝑖2 + 𝜎ˆ𝑘2 ) 2

!

Substituting this into the gradient expression yields: 2 2 2 Δ𝑖,𝑘 𝑤 (𝑦𝑖 ) 1 𝜎ˆ𝑖 + 𝜎ˆ𝑘 𝑐𝑖,𝑘 𝐹 (𝑖, 𝑘) = exp + + ln 2 2 𝜏𝑍𝑖 2𝜎ˆ𝑖 𝜎ˆ𝑘 𝜏 4(𝜎ˆ𝑖 + 𝜎ˆ𝑘 ) 2

!

Finally, we perform the asymptotic analysis for the topological hallucination scenario. Comparing 𝐹 (𝑖, 𝑘) with an ordinary hard negative 𝐹 (𝑖, 𝑚), the partition function 𝑍𝑖 and base weight 𝑤 (𝑦𝑖 ) cancel out: 𝑐 − 𝑐  𝐹 (𝑖, 𝑘) BC(𝑖, 𝑚) 𝑖,𝑚 𝑖,𝑘 = exp 𝐹 (𝑖, 𝑚) BC(𝑖, 𝑘) 𝜏 ! 2 2 Δ𝑖,𝑘 Δ𝑖,𝑚 𝑐𝑖,𝑘 − 𝑐𝑖,𝑚 = exp − + + 𝐶𝑖,𝑘,𝑚 2) 𝜏 4(𝜎ˆ𝑖2 + 𝜎ˆ𝑘2 ) 4(𝜎ˆ𝑖2 + 𝜎ˆ𝑚 (𝜎ˆ 2 +𝜎ˆ 2 ) 𝜎ˆ𝑚

where 𝐶𝑖,𝑘,𝑚 = 12 ln (𝜎ˆ𝑖2 +𝜎ˆ𝑘2 ) 𝜎ˆ is the logarithmic variance term. Given 𝜎ˆ𝑖 , 𝜎ˆ𝑘 , 𝜎ˆ𝑚 = O (1) and 𝑐𝑖,𝑘 , 𝑐𝑖,𝑚 ∈ [−1, 1], 𝑖

𝑚

𝑘

both 𝐶𝑖,𝑘,𝑚 and the similarity residual are bounded. As Δ𝑖,𝑘 → ∞, the squared-exponential term dominates. Letting

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Zhou et al.

Σ2 = 4(𝜎ˆ𝑖2 + 𝜎ˆ𝑘2 ), we obtain the asymptotic equivalence: 2 Δ𝑖,𝑘 𝐹 (𝑖, 𝑘) ∼ exp 𝐹 (𝑖, 𝑚) Σ2

! → +∞ □

The derivation above highlights two key properties of our weighting mechanism: • Correction of Topological Errors: When samples with large semantic gaps are erroneously clustered (𝑐𝑖,𝑘 → 1 and Δ𝑖,𝑘 → ∞), the squared-exponential repulsive gradient provides a sufficiently large penalty to rectify the distorted feature manifold. • Uncertainty-Adaptivity: The growth of the repulsive gradient is constrained by Σ2 in the denominator, which is inversely proportional to the predicted variances. For noisy samples with high uncertainty (large variance), gradient amplification is naturally suppressed, preventing gradient explosion and maintaining training stability. F

Experiment Details

To comprehensively evaluate DUO, we compare it against representative deep imbalanced regression (DIR) baselines: Vanilla relies solely on the standard regression loss; LDS & FDS (Yang et al. [8]) leverage the neighboring correlation of continuous targets to jointly smooth and calibrate both target and feature distributions; RankSim (Gong et al. [2]) introduces ranking regularization, forcing the neighbor topology in the feature space to strictly align with the distance ranking in the target space; Balanced MSE (Ren et al., 2022) incorporates target priors, modifying the mean squared error from a statistical perspective to recover unbiased predictions; ConR (Keramati et al. [5]) models target similarities via contrastive learning to prevent minority features from collapsing into majority ones. We also include the recent Dist Loss (Nie et al. [6]), which explicitly aligns the predicted distribution with the ground-truth target distribution by jointly penalizing sample-level errors and macroscopic distribution distances. All models are trained using a single NVIDIA GeForce RTX 4090 D GPU. To ensure fair comparisons, all standard training, validation, and testing splits strictly follow the configurations set by Yang et al.[8]. The remainder of this section provides implementation details and hyper-parameter selections for the three datasets. For the imbalance-aware weight, we discretize the continuous label space into bins using the training set and define 𝑤 (𝑦𝑖 ) as the inverse empirical frequency of the bin containing 𝑦𝑖 . This global scarcity factor is multiplied by the detached instance-level uncertainty in L𝑀𝑒𝑎𝑛 . Training uses the squared-error mean objective, whereas MAE, bMAE, and GM are evaluation metrics. F.1

Age estimation

For the AgeDB-DIR and IMDB-WIKI-DIR benchmarks, we adopt ResNet-50 as the backbone encoder, followed by independent fully connected heads to output 𝜇 and 𝜎, respectively. The batch size is set to 64, and the initial learning rate is 2.5 × 10−4 , which decays by a factor of 10 at epoch 60 and epoch 80. We use the Adam optimizer (momentum of 0.9, weight decay of 10−4 ) and Mean Squared Error (MSE) as the base regression loss. All models are trained for 120 epochs. The input image size is 224x224, and data augmentation includes random cropping and random horizontal flipping. Label smoothing is applied using a Gaussian kernel (kernel size 9, 𝜎lds = 1), and the re-weighting scheme uses inverse frequency weighting. The hyper-parameters for DUO are set as follows: variance term weight 𝛽 = 1.0, alignment weight 𝜆align = 1.0, temperature parameter 𝜏 = 0.07, Gaussian overlap threshold 𝛿 = 0.5, and push coefficient

Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 9 𝜂 = 0.01. The number of warm-up epochs is 15, validation MAE is used as the monitoring metric, and the random seed is fixed to 42. F.2

Depth estimation

For the NYUD2-DIR benchmark, we employ a ResNet-50 based encoder-decoder architecture (Hu et al.[3]), outputting depth maps with a resolution of 152x114. Unlike age estimation, the prediction targets are dense depth maps; thus, 𝝁 and 𝝈 are output pixel-wise. The loss function terms are computed pixel-wise within the valid pixel mask and then averaged. In the contrastive alignment module, the sample-level proxy target 𝑦¯𝑖 is defined as the spatial mean of the depth map over valid pixels, and 𝜎¯𝑖 is computed similarly, thereby constructing the Gaussian distribution N (𝑦¯𝑖 , 𝜎¯𝑖 ) for positive and negative sample partitioning. Other training configurations include: batch size of 12, initial learning rate of 2.5 × 10−4 (decaying by a factor of 10 every 5 epochs), Adam optimizer (momentum 0.9, weight decay 10−4 ), MSE regression loss, and 120 training epochs. Data augmentation includes random horizontal flipping, ±5° rotation, color jittering, and PCA lighting perturbations. Label smoothing utilizes a Gaussian kernel (kernel size 9, 𝜎lds = 1) with inverse frequency re-weighting. DUO hyper-parameters: 𝛽 = 1.0, 𝜆align = 0.1, 𝜏 = 0.07, 𝛿 = 0.5, 𝜂 = 0.01, 10 warm-up epochs, and random seed 42. F.3

Protein estimation

For the protein fitness dataset, we use AAV2-DIR, which contains 42,329 AAV2 capsid protein variant sequences and their fitness scores. These are randomly split into training, validation, and testing sets at a ratio of 7:1.5:1.5. Each sequence is embedded into a 1024-dimensional vector by mean-pooling ProtBERT residue embeddings from ProtTrans[1]. The downstream model is a two-layer Transformer encoder (𝑑 model = 256, 8 attention heads, FFN dimension 512), which splits the 1024-dimensional input into 64 patches followed by sequence mean pooling. The DUO branch outputs 𝜇 and 𝜎. Training configurations: batch size of 128, Adam optimizer, initial learning rate of 2.5 × 10−4 , MSE loss, and a maximum of 200 epochs. DUO hyper-parameters: 𝛽 = 1.0, 𝜆align = 1.0, 𝜏 = 0.07, 𝛿 = 0.5, 10 warm-up epochs, inverse frequency re-weighting, and random seed 42. F.4

Additional Analyses from the Rebuttal

Controlled component ablation. Table S1 isolates four factors under matched dual-head capacity: architectural capacity, mean–variance decoupling, uncertainty weighting, and the contrastive assignment rule. Dual-head controls share DUO’s exact prediction-head structure; Vanilla is retained as the single-head reference. Pure dual-head NLL fails to improve over single-head Vanilla, confirming that gains do not arise from extra parameters. Each removal or replacement of a DUO component—disabling decoupling, disabling dynamic weighting, or replacing BC with ConR—consistently degrades few-shot metrics, confirming that all three components contribute to tail performance. Uncertainty and overlap controls. Replacing learned uncertainty with the instantaneous absolute residual removes the variance head and degrades most tail metrics, indicating that the learned predictor provides a smoother difficulty estimate than a batch-wise residual. Replacing BC with interval IoU, which reduces each Gaussian to a rigid interval, also performs worse than the full model. Feature-space smoothness. Because t-SNE is qualitative, we additionally measure local smoothness on AgeDB-DIR using KNN-MAE (𝑘 = 5), which computes the average label discrepancy between each sample and its 𝑘 nearest

10

Zhou et al. Table S1. Fine-grained ablation in the few-shot regions.

Method

Head

Vanilla NLL NLL + BC DUO (NoWeight) DUO + ConR DUO

Single Dual Dual Dual Dual Dual

Key setting

MAE ↓

Standard MSE Joint mean–variance Joint NLL + BC Decoupled + BC Decoupled + ConR Decoupled + weight + BC

bMAE ↓

GM ↓

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

25.546 25.842 24.295 24.560 23.996 22.940

13.381 14.280 11.916 12.226 11.565 9.917

7.365 7.880 4.160 4.109 4.208 4.106

33.085 33.986 31.334 31.821 31.269 27.190

16.579 16.710 14.347 14.968 14.265 11.590

7.565 8.095 4.396 4.380 4.596 4.329

18.091 17.862 16.775 16.306 15.633 13.650

9.644 15.850 8.762 8.447 7.996 6.587

7.351 8.274 3.823 3.815 3.869 3.777

Table S2. Few-shot controls on uncertainty representation and overlap metric.

MAE ↓

bMAE ↓

GM ↓

Method

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

IMDB

AgeDB

AAV2

DUO Raw residual Interval IoU

22.94 24.37 23.80

9.92 12.00 11.40

4.11 4.15 4.20

27.19 30.75 30.90

11.59 13.61 13.80

4.33 4.39 4.50

13.65 16.61 15.10

6.59 8.65 7.60

3.78 3.82 3.90

neighbors in feature space. DUO yields the lowest neighborhood error in both the compound tail and median-shot regions, supporting the local-continuity interpretation of the visualization. Table S3. KNN-MAE feature smoothness on AgeDB-DIR. Lower is better.

Method

Few ∪ Old

Median

Few-shot MAE

Vanilla RankSim FDS DUO

7.51 7.90 7.15 7.03

4.88 5.48 4.97 4.61

13.38 14.60 12.14 11.79

Warm-up sensitivity. Table S4 shows a stable U-shaped trend around the selected 𝑇𝑤 = 15 on AgeDB-DIR, rather than sensitivity to a single unstable setting. Table S4. Sensitivity to the warm-up epoch 𝑇𝑤 on AgeDB-DIR.

𝑇𝑤

MAE ↓

bMAE ↓

GM ↓

5 15 20 50

11.324 9.917 10.602 10.702

14.012 11.590 13.102 13.202

7.801 6.587 7.302 7.352

Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 11 Table S5. Experimental results on the IMDB-WIKI dataset across different shot regions (Overall, Few, Median, Many). Overall

Few

Median

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

Vanilla NLL LDS FDS RankSim ConR Balanced MSE DistLoss DUO

7.829 8.094 7.626 7.829 7.892 7.817 7.943 11.725 8.420

13.717 14.201 12.739 13.294 13.708 13.797 12.924 15.966 12.850

4.389 4.519 4.257 4.457 4.443 4.430 4.639 7.368 4.580

25.546 25.842 22.755 24.347 25.317 25.969 23.541 24.215 22.941

33.085 33.986 30.179 31.839 32.464 33.601 30.000 30.165 27.190

18.091 17.862 13.786 14.600 17.639 17.990 13.803 14.076 13.650

14.387 15.221 12.146 12.762 14.914 14.323 12.258 16.316 14.120

14.933 15.918 12.597 13.335 15.546 14.836 12.679 16.450 14.350

9.797 10.420 6.938 7.145 10.401 9.859 7.054 10.145 9.050

7.052 7.266 7.059 7.209 7.077 7.039 7.388 11.179 7.750

7.146 7.371 7.133 7.283 7.181 7.130 7.461 11.252 7.830

4.024 4.133 4.024 4.219 4.058 4.064 4.416 7.111 4.380

Table S6. Experimental results on the AgeDB dataset across different shot regions (Overall, Few, Median, Many). Overall

Few

Median

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

Vanilla NLL LDS FDS RankSim ConR DistLoss Balanced MSE DUO

7.754 8.220 7.904 7.679 8.320 7.345 10.017 8.190 7.452

9.848 10.000 8.826 9.574 10.504 9.018 10.825 9.070 8.468

4.951 8.990 5.139 4.906 5.331 4.681 6.316 8.420 4.799

13.381 14.280 11.489 12.136 14.601 12.333 12.121 9.690 9.917

16.579 16.710 12.055 15.531 17.611 14.486 13.670 11.940 11.590

9.644 15.850 7.577 8.326 11.021 9.376 7.726 10.480 6.587

9.172 9.980 8.435 8.956 10.069 8.482 10.300 7.640 8.305

9.230 9.990 8.426 9.000 10.114 8.550 10.308 7.640 8.335

6.175 9.850 5.404 5.811 7.042 5.845 6.531 7.490 5.162

6.743 6.950 7.369 6.833 7.143 6.484 9.645 8.080 6.980

6.743 6.950 7.369 6.833 7.143 6.484 9.645 8.080 6.980

4.324 6.810 4.860 4.415 4.550 4.075 6.087 7.820 4.509

Table S7. Experimental results on the AAV2 dataset across different shot regions (Overall, Few, Median, Many). Overall

Few

Median

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

Vanilla LDS FDS RankSim ConR Balanced MSE DistLoss DUO

2.643 1.990 1.988 1.936 1.719 2.650 2.652 1.672

2.623 1.981 1.985 1.932 1.716 2.639 2.624 1.669

2.079 1.178 1.256 1.189 0.992 2.115 2.013 1.023

7.365 4.361 5.023 4.885 4.183 7.283 7.432 4.106

7.565 4.681 5.388 5.225 4.305 7.563 7.567 4.329

7.351 4.054 4.786 4.652 3.932 7.253 7.416 3.777

5.918 3.648 4.345 4.151 3.358 5.992 5.856 3.231

6.116 3.888 4.667 4.408 3.590 6.226 6.024 3.437

5.890 3.211 4.008 3.857 3.043 5.978 5.813 2.855

2.534 1.935 1.911 1.864 1.665 2.539 2.545 1.620

2.757 2.117 2.147 2.110 1.852 2.745 2.788 1.774

2.010 1.140 1.210 1.145 0.957 2.045 1.945 0.989

Table S8. Experimental results on the NYUD2 dataset across different shot regions (Overall, Few, Median, Many). Overall

Few

Median

Many

Method

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

Vanilla LDS FDS RankSim ConR DUO

0.580 0.633 0.589 0.594 0.586 0.597

0.371 0.434 0.374 0.377 0.378 0.396

1.051 1.004 1.067 1.066 1.067 0.990

0.807 0.737 0.807 0.803 0.801 0.781

1.867 1.764 1.905 1.896 1.869 1.759

1.443 1.333 1.478 1.475 1.459 1.351

1.884 1.714 1.920 1.907 1.912 1.696

0.608 0.641 0.594 0.601 0.591 0.643

0.795 0.781 0.802 0.818 0.805 0.777

0.596 0.590 0.601 0.622 0.601 0.586

0.673 0.658 0.675 0.698 0.683 0.653

0.668 0.680 0.661 0.638 0.668 0.668

0.457 0.546 0.464 0.470 0.463 0.497

0.318 0.392 0.320 0.322 0.325 0.350

0.363 0.426 0.364 0.367 0.370 0.423

0.824 0.744 0.825 0.822 0.818 0.795

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Zhou et al.

Table S9. Pairwise comparison of various baselines and their DUO-enhanced counterparts on the AgeDB dataset across different shot regions. Overall

Few

Median

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

LDS LDS+DUO

7.904 8.121

8.826 8.983

5.139 5.116

11.489 9.595

12.055 11.563

7.577 5.415

8.435 9.411

8.426 9.396

5.404 5.860

7.369 7.588

7.369 7.588

4.860 4.887

FDS FDS+DUO

7.679 7.699

9.574 9.137

4.906 4.892

12.136 11.561

15.531 13.751

8.326 8.267

8.956 9.028

9.000 9.054

5.811 5.929

6.833 6.901

6.833 6.901

4.415 4.374

RankSim RankSim+DUO

8.320 8.346

10.504 9.525

5.330 5.478

14.601 11.640

17.611 13.334

11.021 7.124

10.069 9.368

10.114 9.393

7.042 6.302

7.143 7.698

7.143 7.698

4.550 5.114

Balanced MSE Balanced MSE+DUO

8.190 6.660

9.070 7.400

8.420 6.840

9.690 9.120

11.940 10.160

10.480 9.510

7.640 8.600

7.640 8.600

7.490 8.560

8.080 5.880

8.080 5.880

7.820 5.610

Table S10. Pairwise comparison of various baselines and their DUO-enhanced counterparts on the IMDB-WIKI dataset across different shot regions. Overall

Few

Median

Many

Method

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

MAE ↓

bMAE ↓

GM ↓

LDS LDS+DUO

7.626 8.492

12.739 13.848

4.257 5.038

22.755 24.378

30.179 31.551

13.786 16.595

12.146 14.398

12.597 14.793

6.938 9.392

7.059 7.793

7.133 7.885

4.024 4.697

FDS FDS+DUO

7.829 8.404

13.294 13.815

4.457 4.893

24.347 24.471

31.839 31.432

14.600 16.903

12.762 14.743

13.335 15.256

7.145 9.874

7.209 7.664

7.283 7.755

4.219 4.535

RankSim RankSim+DUO

7.892 8.458

13.708 13.732

4.443 4.953

25.317 23.911

32.464 30.774

17.639 15.731

14.914 14.923

15.546 15.372

10.401 9.955

7.077 7.714

7.181 7.809

4.058 4.596

Table S11. Pairwise comparison of various baselines and their DUO-enhanced counterparts on the NYUD2 dataset across different shot regions. Overall

Few

Median

Many

Method

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

RMSE ↓

MAE ↓

bMAE ↓

𝛿1 ↑

LDS LDS+DUO

0.633 0.626

0.434 0.417

1.004 1.008

0.737 0.762

1.764 1.767

1.333 1.341

1.714 1.727

0.641 0.648

0.781 0.821

0.590 0.624

0.658 0.684

0.680 0.651

0.546 0.528

0.392 0.370

0.426 0.415

0.744 0.774

FDS FDS+DUO

0.589 0.607

0.374 0.396

1.067 1.054

0.807 0.783

1.905 1.845

1.478 1.443

1.920 1.854

0.594 0.606

0.802 0.819

0.601 0.621

0.675 0.700

0.661 0.660

0.464 0.494

0.320 0.344

0.364 0.391

0.825 0.799

RankSim RankSim+DUO

0.594 0.604

0.377 0.393

1.066 1.066

0.803 0.789

1.896 1.852

1.475 1.455

1.907 1.882

0.601 0.590

0.818 0.827

0.622 0.622

0.698 0.710

0.638 0.660

0.470 0.488

0.322 0.340

0.367 0.387

0.822 0.805

Supplementary Material for “Beyond Homoscedasticity: Decoupled Uncertainty Optimization for Deep Imbalanced Regression” 13 References [1] Ahmed Elnaggar, Michael Heinzinger, Christian Dallago, Ghalia Rehawi, Yu Wang, Llion Jones, Tom Gibbs, Tamas Feher, Christoph Angerer, Martin Steinegger, Debsindhu Bhowmik, and Burkhard Rost. 2022. ProtTrans: Toward understanding the language of life through self-supervised learning. IEEE Transactions on Pattern Analysis and Machine Intelligence 44, 10 (2022), 7112–7127. [2] Yu Gong, Greg Mori, and Frederick Tung. 2022. Ranksim: Ranking similarity regularization for deep imbalanced regression. In Proceedings of the 39th International Conference on Machine Learning (Proceedings of Machine Learning Research, Vol. 162). PMLR, 7634–7649. [3] Junjie Hu, Mete Ozay, Yan Zhang, and Takayuki Okatani. 2019. Revisiting single image depth estimation: Toward higher resolution maps with accurate object boundaries. In 2019 IEEE winter conference on applications of computer vision (WACV). IEEE, 1043–1051. [4] Edwin T Jaynes. 1957. Information theory and statistical mechanics. Physical review 106, 4 (1957), 620–630. [5] Mahsa Keramati, Lili Meng, and R David Evans. 2024. Conr: Contrastive regularizer for deep imbalanced regression. In International Conference on Learning Representations. [6] Guangkun Nie, Gongzheng Tang, and Shenda Hong. 2025. Dist loss: enhancing regression in few-shot region through distribution distance constraint. In International Conference on Learning Representations. [7] David A Nix and Andreas S Weigend. 1994. Estimating the mean and variance of the target probability distribution. In Proceedings of 1994 ieee international conference on neural networks (ICNN’94), Vol. 1. IEEE, 55–60. [8] Yuzhe Yang, Kaiwen Zha, Yingcong Chen, Hao Wang, and Dina Katabi. 2021. Delving into deep imbalanced regression. In Proceedings of the 38th International Conference on Machine Learning (Proceedings of Machine Learning Research, Vol. 139). PMLR, 11842–11851.